Congruency-Constrained TU Problems Beyond the Bimodular Case
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A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs min{c^⊤ x Tx≤ b, γ^⊤ x≡ r*m, x∈ℤ^n} with a totally unimodular constraint matrix T. Such problems have been shown to be polynomial-time solvable for m=2, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose n× n subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., for m>2. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m=3. Furthermore, for general m, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation.
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